Lie algebra from Dynkin diagram

Here is the construction I gave in class for a Lie algebra given its Dynkin diagram.

Next will be some stuff about Weyl groups and Weyl chambers, delaying the fact that simple reflections generate N(T)/T.

We're going to do other chapters from here next, though in a different order: first the Bruhat decomposition, then stuff about Weyl chambers, filling the hole mentioned above, and then the Borel-Weil theorem that lets us construct all the irreps for complex reductive Lie groups.

April 14

If K is compact with discrete center, pi_1(K) is finite. Hence there are finitely
many Lie groups with that Lie algebra. Proof: use Hurewicz to connect to H^1, and Lie algebra cohomology to show that H^1 vanishes over R.

The angle between any two roots must be 90, 120, 135, 150, or 180.

There are at most two root lengths in a connected root system, and if two they differ by a factor of sqrt(2) or sqrt(3).

Next time: simple systems, Dynkin diagrams, their classification.

Old notes

Here are my notes from a previous incarnation of this class. The bottom one is the one we followed to get to root systems.